BGG-type relations for transfer matrices of rational spin chains and the shifted Yangians

In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R.Frassek, I. Karpov, and V.Pestun.